Numerical Methods for Large Eigenvalue Problems: Revised Edition

The Lanczos algorithm for extensive many-body systems in the thermodynamic limit

Witte, N. S., & Bessis, D. (1999) · Journal of Mathematical Physics · doi:10.1063/1.533010 · Cited by 11

A positivity preserving inexact Noda iteration for computing the smallest eigenpair of a large irreducible $$M$$ M -matrix

Jia, Z., Lin, W., & Liu, C.-S. (2014) · Numerische Mathematik · doi:10.1007/s00211-014-0677-2 · Cited by 11

ARNOLDI-FABER METHOD FOR LARGE NON HERMITIAN EIGENVALUE PROBLEMS

Heuveline, V., & Sadkane, M. (1997) · Cited by 11

Accelerating Convergence by Augmented Rayleigh--Ritz Projections For Large-Scale Eigenpair Computation

Wen, Z., & Zhang, Y. (2017) · SIAM Journal on Matrix Analysis and Applications · doi:10.1137/16m1058534 · Cited by 11

Sampling and multilevel coarsening algorithms for fast matrix approximations

Ubaru, S., & Saad, Y. (2019) · Numerical Linear Algebra with Applications · doi:10.1002/nla.2234 · Cited by 10

Leader‐following consensus of multiple uncertain Euler‐Lagrange systems

Lu, M., & Liu, L. (2018) · International Journal of Robust and Nonlinear Control · doi:10.1002/rnc.4223 · Cited by 11

pizza: an open-source pseudo-spectral code for spherical quasi-geostrophic convection

Gastine, T. (2019) · Geophysical Journal International · doi:10.1093/gji/ggz103 · Cited by 12

Inexact Newton Methods Applied to Under-Determined Systems

Simonis, J. P. (2006) · Digital WPI · Cited by 11

Spatial Modulation for Uplink Multi-User mmWave MIMO Systems With Hybrid Structure

Wang, W., & Zhang, W. (2019) · IEEE Transactions on Communications · doi:10.1109/tcomm.2019.2948340 · Cited by 11

Numerical solution of two-energy-group neutron noise diffusion problems with fine spatial meshes

Mylonakis, A. G., Vinai, P., & Demazière, C. (2019) · Annals of Nuclear Energy · doi:10.1016/j.anucene.2019.107093 · Cited by 11

Tensor Network States for Vibrational Spectroscopy

Glaser, N., Baiardi, A., & Reiher, M. (2022) · WORLD SCIENTIFIC eBooks · doi:10.1142/9789811237911_0003 · Cited by 11

Solution of nonlinear time-dependent PDEs through componentwise approximation of matrix functions

Cibotarica, A., Lambers, J. V., & Palchak, E. M. (2016) · Journal of Computational Physics · doi:10.1016/j.jcp.2016.06.024 · Cited by 11

Quantum algorithms for SVD-based data representation and analysis

Bellante, A., Luongo, A., & Zanero, S. (2022) · Quantum Machine Intelligence · doi:10.1007/s42484-022-00076-y · Cited by 16

Eigenstructure of rank one updated matrices

Bru, R., Cantó, R., & Urbano, A. M. (2015) · Linear Algebra and its Applications · doi:10.1016/j.laa.2015.07.036 · Cited by 11

A refined jacobi-davidson method and its correction equation

Feng, S., & Jia, Z. (2005) · Computers & Mathematics with Applications · doi:10.1016/j.camwa.2003.01.018 · Cited by 11

Continuous methods for symmetric generalized eigenvalue problems

Gao, X., Golub, G. H., & Liao, L. (2007) · Linear Algebra and its Applications · doi:10.1016/j.laa.2007.08.034 · Cited by 11

Parallel computation of pseudospectra of large sparse matrices

Mezher, D., & Philippe, B. (2002) · Parallel Computing · doi:10.1016/s0167-8191(01)00136-3 · Cited by 11

Iterative Validation of Eigensolvers: A Scheme for Improving the Reliability of Hermitian Eigenvalue Solvers

McCombs, J. R., & Stathopoulos, A. (2006) · SIAM Journal on Scientific Computing · doi:10.1137/050627617 · Cited by 11

Ritz and Pseudo-Ritz values using matrix polynomials

Simoncini, V. (1996) · Linear Algebra and its Applications · doi:10.1016/0024-3795(95)00682-6 · Cited by 11

A Stratification of the Set of Normal Matrices

Huhtanen, M. (2001) · SIAM Journal on Matrix Analysis and Applications · doi:10.1137/s0895479899357085 · Cited by 11

Locking issues for finding a large number of eigenvectors of Hermitian matrices

Stathopoulos, A. (2006) · Cited by 11

Harmonic and refined harmonic shift-invert residual Arnoldi and Jacobi–Davidson methods for interior eigenvalue problems

Jia, Z., & Li, C. (2015) · Journal of Computational and Applied Mathematics · doi:10.1016/j.cam.2014.12.043 · Cited by 11

Algorithms and perturbation theory for matrix eigenvalue problems and the singular value decomposition

Freund, R. W., & Nakatsukasa, Y. (2011) · Cited by 11

Transverse Combustion Instabilities: Modern Experimental Techniques and Analysis

Quinlan, J. M., & Zinn, B. T. (2014) · AIAA/ASME/SAE/ASEE Joint Propulsion Conference (JPC) · doi:10.2514/6.2014-3682 · Cited by 11

Projected nonmonotone search methods for optimization with orthogonality constraints

Dalmau, O., & Oviedo, H. (2017) · Computational and Applied Mathematics · doi:10.1007/s40314-017-0501-6 · Cited by 11

A Survey of Componentwise Perturbation Theory

Higham, N. J. (1994) · MIMS EPrints (University of Southampton) · Cited by 11

PAT – a Reliable Path-Following Algorithm

Mezher, D., & Philippe, B. (2002) · Numerical Algorithms · doi:10.1023/a:1014824425949 · Cited by 11

The methods of Vorobyev and Lanczos

Brezinski, C. (1996) · Linear Algebra and its Applications · doi:10.1016/0024-3795(94)00081-6 · Cited by 11

Fast, scalable master equation solution algorithms. IV. Lanczos iteration with diffusion approximation preconditioned iterative inversion

Frankcombe, T. J., & Smith, S. C. (2003) · The Journal of Chemical Physics · doi:10.1063/1.1628214 · Cited by 11

BWR stability monitoring using adaptive methods

Navarro-Esbrí, J., Ginestar, D., & Verdú, G. (2003) · Annals of Nuclear Energy · doi:10.1016/s0306-4549(02)00145-7 · Cited by 11

Application of the Arnoldi method in FEM analysis of waveguides

Mielewski, J., & Mrozowski, M. (1998) · IEEE Microwave and Guided Wave Letters · doi:10.1109/75.650971 · Cited by 11

Accelerated FD analysis of dielectric resonators

Mielewski, J., Cwikla, A., & Mrozowski, M. (1998) · IEEE Microwave and Guided Wave Letters · doi:10.1109/75.736249 · Cited by 11

A block Rayleigh quotient iteration with local quadratic convergence.

Fattebert, J. (1998) · Cited by 11

Ab Initio Simulations of Light Nuclear Systems Using Eigenvector Continuation and Auxiliary Field Monte CarloPreprint

Frame, D. (2019) · arXiv (Cornell University) · doi:10.48550/arxiv.1905.02782 · Cited by 11

Dynamic bounds on stochastic chemical kinetic systems using semidefinite programmingPreprint

Dowdy, G. R., & Barton, P. I. (2018) · DSpace@MIT (Massachusetts Institute of Technology) · Cited by 11

Limited Memory Block Krylov Subspace Optimization for Computing Dominant Singular Value Decompositions

Liu, X., Wen, Z., & Zhang, Y. (2012) · doi:10.21236/ada580501 · Cited by 11

On Chebyshev–Davidson Method for Symmetric Generalized Eigenvalue Problems

Miao, C.-Q. (2020) · Journal of Scientific Computing · doi:10.1007/s10915-020-01360-4 · Cited by 10

Efficient Semidefinite Programming with Approximate ADMM

Rontsis, N., Goulart, P. J., & Nakatsukasa, Y. (2021) · Journal of Optimization Theory and Applications · doi:10.1007/s10957-021-01971-3 · Cited by 12

Linear global and asymptotic stability analysis of the flow past rectangular cylinders moving along a wall

Chiarini, A., & Auteri, F. (2023) · Journal of Fluid Mechanics · doi:10.1017/jfm.2023.434 · Cited by 10

Quasiperiodic perturbations of Stokes waves: Secondary bifurcations and stability

Dyachenko, S. A., & Semenova, A. (2023) · Journal of Computational Physics · doi:10.1016/j.jcp.2023.112411 · Cited by 10

Efficient uncertainty propagation for stochastic multiscale linear elasticity

Zheng, Z., & Nackenhorst, U. (2024) · Computer Methods in Applied Mechanics and Engineering · doi:10.1016/j.cma.2024.117085 · Cited by 10

A theoretical analysis of mass scaling techniques

Voet, Y., Sande, E., & Buffa, A. (2025) · Computational Mechanics · doi:10.1007/s00466-025-02611-7 · Cited by 11

The Modal Substructuring Method: An Efficient Technique for Large-Size Numerical Simulations

Laffay, P. O., Quéméner, O., Neveu, A., et al. (2011) · Numerical Heat Transfer Part B Fundamentals · doi:10.1080/10407790.2011.609113 · Cited by 10

Dissociative ionization of H2+ : Few-cycle effect in the joint electron-ion energy spectrum

Mosert, V., & Bauer, D. (2015) · Physical Review A · doi:10.1103/physreva.92.043414 · Cited by 10

The inexact residual iteration method for quadratic eigenvalue problem and the analysis of convergence

Liu, Y., Sun, Y., & Gong, F. (2017) · Journal of Computational and Applied Mathematics · doi:10.1016/j.cam.2017.10.003 · Cited by 10

Spatially Sparse Beamforming Training for Millimeter Wave MIMO Systems

Cheng, X., Lou, N., & Li, S. (2017) · IEEE Transactions on Wireless Communications · doi:10.1109/twc.2017.2682167 · Cited by 10

Floquet analysis of spatially periodic thermocapillary convection in a low-Prandtl-number liquid bridge

Motegi, K., Fujimura, K., & Ueno, I. (2017) · Physics of Fluids · doi:10.1063/1.4993466 · Cited by 10

What to expect after the honeymoon: evolutionary psychology of part-time franchising

Thaichon, P., Weaven, S. K., Quach, S., Baker, B. L., & Frazer, L. (2019) · Journal of Strategic Marketing · doi:10.1080/0965254x.2019.1570315 · Cited by 10

An almost subharmonic instability in the flow past rectangular cylinders

Chiarini, A., Quadrio, M., & Auteri, F. (2022) · Journal of Fluid Mechanics · doi:10.1017/jfm.2022.712 · Cited by 11

Estimating Eigenenergies from Quantum Dynamics: A Unified Noise-Resilient Measurement-Driven Approach

Shen, Y., Camps, D., Szasz, A., Darbha, S., Klymko, K., Williams‐Young, D. B., Tubman, N. M., & Beeumen, R. V. (2025) · Quantum · doi:10.22331/q-2025-08-27-1836 · Cited by 17

On the efficient numerical solution of lattice systems with low-order couplings

Ammon, A., Genz, A. C., Hartung, T., et al. (2015) · Computer Physics Communications · doi:10.1016/j.cpc.2015.09.004 · Cited by 10

An analogue of the results of Saad and Stewart for harmonic Ritz vectors

Chen, G., & Jia, Z. (2004) · Journal of Computational and Applied Mathematics · doi:10.1016/j.cam.2003.10.020 · Cited by 10

Ab initio calculation of the deformation potential and photoelastic coefficients of silicon with a non-uniform finite-difference solver based on the local density approximation

Witzens, J. (2014) · Computer Physics Communications · doi:10.1016/j.cpc.2014.03.016 · Cited by 10

Acoustic resonances in a 3D open cavity with non-parallel walls

Ortiz, S., González, L. M., Díaz, C. G., et al. (2015) · Journal of Sound and Vibration · doi:10.1016/j.jsv.2015.11.013 · Cited by 10

On Solving Ill Conditioned Linear Systems

Douglas, C. C., Lee, L., & Yeung, M.-C. (2016) · Procedia Computer Science · doi:10.1016/j.procs.2016.05.386 · Cited by 10

A Jacobi-Davidson Iteration Method for Linear Eigenvalue

Sleijpen, G. L. G., & van der Vorst, H. A. (2000) · Cited by 10

A rational deferred correction approach to parabolic optimal control problems

Güttel, S., & Pearson, J. W. (2017) · IMA Journal of Numerical Analysis · doi:10.1093/imanum/drx046 · Cited by 10

Eigenvalue Solvers for Modeling Nuclear Reactors on Leadership Class Machines

Slaybaugh, R. N., Ramirez-Zweiger, M., Pandya, T. M., et al. (2018) · Nuclear Science and Engineering · doi:10.1080/00295639.2017.1413875 · Cited by 10

Similarity-Aware Spectral Sparsification by Edge Filtering

Feng, Z. (2018) · Proceedings - ACM IEEE Design Automation Conference · doi:10.1109/dac.2018.8465798 · Cited by 10

Filtered Krylov-like sequence method for symmetric eigenvalue problems

Miao, C.-Q. (2018) · Numerical Algorithms · doi:10.1007/s11075-018-0627-7 · Cited by 10

On choices of formulations of computing the generalized singular value decomposition of a large matrix pair

Huang, J., & Jia, Z. (2020) · Numerical Algorithms · doi:10.1007/s11075-020-00984-9 · Cited by 10

A parallel generalized conjugate gradient method for large scale eigenvalue problems

Li, Y., Xie, H., Xu, R., et al. (2020) · CCF Transactions on High Performance Computing · doi:10.1007/s42514-020-00029-6 · Cited by 10

Optimization of flux mapping in-core detector locations in AHWR using clustering approach

Anupreethi, B., Gupta, A., Kannan, U., et al. (2020) · Nuclear Engineering and Design · doi:10.1016/j.nucengdes.2020.110756 · Cited by 10

Comparison of Numerical Methods and Open-Source Libraries for Eigenvalue Analysis of Large-Scale Power Systems

Tzounas, G., Dassios, I., Liu, M., & Milano, F. (2020) · Applied Sciences · doi:10.3390/app10217592 · Cited by 10

Simulation of random fields on random domains

Zheng, Z., Valdebenito, M. A., Beer, M., et al. (2023) · Probabilistic Engineering Mechanics · doi:10.1016/j.probengmech.2023.103455 · Cited by 10

Linear stability analysis for a rotating cylinder with a rotating magnetic field

Walker, J. S., & Witkowski, L. M. (2004) · Physics of Fluids · doi:10.1063/1.1737740 · Cited by 10

Direct computation of period doubling bifurcation points of large-scale systems of ODEs using a Newton-Picard method

Engelborghs, K. (1999) · IMA Journal of Numerical Analysis · doi:10.1093/imanum/19.4.525 · Cited by 10

Efficient DC analysis of RVJ circuits for moment and derivative computations of interconnect networks

Batterywala, S. H., & Narayanan, H. (1999) · International Conference on VLSI Design · doi:10.1109/icvd.1999.745144 · Cited by 10

AN ENSEMBLE KALMAN FILTER USING THE CONJUGATE GRADIENT SAMPLER

Bardsley, J. M., Solonen, A., Parker, A. E., et al. (2012) · International Journal for Uncertainty Quantification · doi:10.1615/int.j.uncertaintyquantification.2012003889 · Cited by 10

Global stability analysis method to numerically predict precursor of breakdown voltage

Shibata, H., Ohmichi, Y., Watanabe, Y., et al. (2015) · Plasma Sources Science and Technology · doi:10.1088/0963-0252/24/5/055014 · Cited by 10

Projection methods for large-scale T-Sylvester equations

Dopico, F. M., González, J., Kreßner, D., et al. (2015) · Mathematics of Computation · doi:10.1090/mcom/3081 · Cited by 10

Methods for solution of large optimal control problems that bypass open-loop model reduction

Bewley, T., Luchini, P., & Pralits, J. O. (2016) · Meccanica · doi:10.1007/s11012-016-0547-3 · Cited by 10

Semi-supervised generalized eigenvalues classification

Viola, M., Sangiovanni, M., Toraldo, G., & Guarracino, M. R. (2017) · Annals of Operations Research · doi:10.1007/s10479-017-2674-1 · Cited by 10

Sensitivity of long periodic orbits of chaotic systems

Lasagna, D. (2020) · Physical review. E · doi:10.1103/physreve.102.052220 · Cited by 10

A key for reusable parallel linear algebra software

Noulard, É., & Emad, N. (2001) · Parallel Computing · doi:10.1016/s0167-8191(01)00090-4 · Cited by 10

An inverse iteration method using multigrid for quantum chemistry

Fattebert, J. (1996) · BIT Numerical Mathematics · doi:10.1007/bf01731930 · Cited by 10

Adaptive multilevel finite element solution of the Poisson-Boltzmann equation I. Algorithms and examples

Holst, M., Baker, N., & Wang, F. (2000) · Journal of Computational Chemistry · doi:10.1002/1096-987x(20001130)21:15 · Cited by 285

Eigenvibrations of a plate with elastically attached loads

Voß, H. (2004) · PubMed · doi:10.15480/882.185 · Cited by 10

Successive eigenvalue relaxation: a new method for the generalized eigenvalue problem and convergence estimates

Ovtchinnikov, E., & Xanthis, L. S. (2001) · Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences · doi:10.1098/rspa.2000.0674 · Cited by 10

Generalized probabilistic principal component analysis of correlated dataPreprint

Gu, M., & Shen, W. (2018) · arXiv (Cornell University) · doi:10.48550/arxiv.1808.10868 · Cited by 10

Randomized Block Gram–Schmidt Process for the Solution of Linear Systems and Eigenvalue Problems

Balabanov, O., & Grigori, L. (2025) · SIAM Journal on Scientific Computing · doi:10.1137/22m1518396 · Cited by 9

A new scaling for the flow instability past symmetric bluff bodies

Chiarini, A., Quadrio, M., & Auteri, F. (2022) · Journal of Fluid Mechanics · doi:10.1017/jfm.2022.99 · Cited by 9

Fourier‐series approach to model order reduction and controller interaction analysis of large‐scale power system models

Rergis, C. M., Betancourt, R., Barocio, E., et al. (2017) · IET Generation Transmission & Distribution · doi:10.1049/iet-gtd.2017.0465 · Cited by 9

Multilevel quasi-Monte Carlo for random elliptic eigenvalue problems II: efficient algorithms and numerical results

Gilbert, A. D., & Scheichl, R. (2023) · IMA Journal of Numerical Analysis · doi:10.1093/imanum/drad009 · Cited by 9

Observability Blocking for Functional Privacy of Linear Dynamic Networks

Zhang, Y., Cheng, R., & Xia, Y. (2023) · Proceedings of the IEEE Conference on Decision & Control, including the Symposium on Adaptive Processes · doi:10.1109/cdc49753.2023.10384211 · Cited by 9

Speeding Up Krylov Subspace Methods for Computing \(\boldsymbol{{f}(A){b}}\) via Randomization

Cortinovis, A., Kreßner, D., & Nakatsukasa, Y. (2024) · SIAM Journal on Matrix Analysis and Applications · doi:10.1137/22m1543458 · Cited by 10

A Communication-Avoiding Thick-Restart Lanczos Method on a Distributed-Memory System

Yamazaki, I., & Wu, K. J. (2012) · Lecture notes in computer science · doi:10.1007/978-3-642-29737-3_39 · Cited by 9

On optimizing Jacobi–Davidson method for calculating eigenvalues in low dimensional structures using eight band k·p model

Andrzejewski, J. (2013) · Journal of Computational Physics · doi:10.1016/j.jcp.2013.04.003 · Cited by 9

Spectral recycling strategies for the solution of nonlinear eigenproblems in thermoacoustics

Salas, P., Giraud, L., Saad, Y. E.-M., et al. (2015) · Numerical Linear Algebra with Applications · doi:10.1002/nla.1995 · Cited by 9

Some properties of LSQR for large sparse linear least squares problems

Jia, Z. (2010) · Journal of Systems Science and Complexity · doi:10.1007/s11424-010-7190-1 · Cited by 9

Efficient algorithm for steady-state stability analysis of large analog/RF circuits

Bolcato, P., Nallatamby, J., Rumolo, C., et al. (2002) · doi:10.1109/mwsym.2001.966928 · Cited by 9

A subspace consensus approach for distributed connectivity assessment of asymmetric networks

Asadi, M. M., Khosravi, M., Blouin, S., et al. (2016) · Proceedings of the IEEE Conference on Decision & Control, including the Symposium on Adaptive Processes · doi:10.1109/cdc.2016.7798921 · Cited by 9

State change modal method for numerical simulation of dynamic processes in a nuclear reactor

Avvakumov, A. V., Strizhov, V. F., Vabishchevich, P. N., et al. (2018) · Progress in Nuclear Energy · doi:10.1016/j.pnucene.2018.02.027 · Cited by 9

Multilevel method to compute the lambda modes of the neutron diffusion equation

Carreño, A., Vidal-Ferràndiz, A., Ginestar, D., et al. (2017) · Applied Mathematics and Nonlinear Sciences · doi:10.21042/amns.2017.1.00019 · Cited by 9

Sparse data formats and efficient numerical methods for uncertainties quantification in numerical aerodynamics

Litvinenko, A., & Matthies, H. G. (2010) · Digitale Bibliothek Braunschweig (Verbundzentrale Göttingen (VZG)) · doi:10.24355/dbbs.084-201011170923-0 · Cited by 9

Exciton mass and exciton spectrum in the cuprous oxide

Alvermann, A., & Fehske, H. (2017) · Journal of Physics B Atomic Molecular and Optical Physics · doi:10.1088/1361-6455/aaa060 · Cited by 9

Gramian‐based model‐order reduction of constrained structural dynamic systems

Uddin, M. M. (2018) · IET Control Theory and Applications · doi:10.1049/iet-cta.2018.5580 · Cited by 9

A restarted Krylov method with inexact inversions

Dax, A. (2018) · Numerical Linear Algebra with Applications · doi:10.1002/nla.2213 · Cited by 9

On multistep Rayleigh quotient iterations for Hermitian eigenvalue problems

Bai, Z., Miao, C.-Q., & Jian, S. (2019) · Computers & Mathematics with Applications · doi:10.1016/j.camwa.2018.12.025 · Cited by 9

Singular perturbation‐based model reduction of power electronic circuits

Khan, H., Bazaz, M. A., & Nahvi, S. A. (2019) · IET Circuits Devices & Systems · doi:10.1049/iet-cds.2018.5234 · Cited by 9

Matrix perturbation theory

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Matrix Computations

Golub, G. (2012) · doi:10.56021/9781421407944 · Cited by 34,505

Matrix Analysis

Horn, R. A., & Johnson, C. R. (1985) · Cited by 22,203

GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems

Saad, Y., & Schultz, M. H. (1986) · SIAM Journal on Scientific and Statistical Computing · doi:10.1137/0907058 · Cited by 11,280

An iteration method for the solution of the eigenvalue problem of linear differential and integral operators

Lanczos, C. (1950) · Journal of research of the National Bureau of Standards · doi:10.6028/jres.045.026 · Cited by 4,356